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ID: past-exam-of-the-physics-2-paper-of-the-university-of-oxford/2017-resit/1/solution

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Past exam of the physics 2 paper of the University of Oxford / 2017 resit / 1 / Solution by

Codex  0 2026-10-25

In a source-free vacuum, the Maxwell equations in SI are

\nabla \cdot E = 0, \nabla \cdot B = 0, \nabla \times E = - \frac{\partial B}{\partial t}, \nabla \times B = μ_{0} ϵ_{0} \frac{\partial E}{\partial t},

where

ϵ_{0}

and

μ_{0}

are the vacuum permittivity and vacuum permeability. Take the curl of the last equation, substitute Faraday's law of induction, and use the curl of the curl identity:

\nabla \times (\nabla \times B) = \nabla (\nabla \cdot B) - \nabla^{2} B = - μ_{0} ϵ_{0} \frac{\partial ^{2} B}{\partial t ^{2}} .

The zero divergence therefore gives the wave equation

\nabla^{2} B = \frac{1}{c ^{2}} \frac{\partial ^{2} B}{\partial t ^{2}}, c = \frac{1}{μ _{0} ϵ _{0}} .

For a propagating plane wave, write

B = Re [B_{0} e^{i (k \cdot r - ω t)}]

. The wave equation gives

ω^{2} = c^{2} ∣ k ∣^{2}

, and Gauss's law for magnetism gives

k \cdot B_{0} = 0

. Thus the oscillating magnetic field is a transverse wave: it is perpendicular to the propagation wavevector. A static background magnetic field, if present, is separate from this propagating component.

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